What is the connection between sine and cosine and geometry?

[aaaa202]

Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x²
And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and so forth.

But for sine and cosine it seems somewhat different if you use the definition with the unit circle. It seems then that all their properties must be proven through geometric arguments. How do you for instance proove that they are defined for all numbers? How do you proove anything about their derivatives and general limits of them, when all you can resort to is a "drawing"?!

Maybe my worries are for nothing, but I still wanted to ask the question and ask whether there exists a definition of them purely in terms of algebraic means.

 

[jedishrfu]

Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x²
And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and so forth.

But for sine and cosine it seems somewhat different if you use the definition with the unit circle. It seems then that all their properties must be proven through geometric arguments. How do you for instance proove that they are defined for all numbers? How do you proove anything about their derivatives and general limits of them, when all you can resort to is a "drawing"?!

Maybe my worries are for nothing, but I still wanted to ask the question and ask whether there exists a definition of them purely in terms of algebraic means.

There are infinite series that are equivalent to the sin and cos functions:

http://en.wikipedia.org/wiki/Trigonometric_functions

look for the series definition a third of the way into the article.

 

[HallsofIvy]

One can also define sine and cosine in terms of an "initial value problem":
y= cos(x) is the function satisfying y''= -y with y(0)= 1, y'(0)= 0.

y= sin(x) is the function satisfying y''= -y with y(0)= 0, y'(0)= 0.

All of the properties can be derived from those. And those facts can be derived from the series definitions jedishrfu cites. Proving periodicity takes some work!

 

[Ray Vickson]

Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x²
And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and so forth.

But for sine and cosine it seems somewhat different if you use the definition with the unit circle. It seems then that all their properties must be proven through geometric arguments. How do you for instance proove that they are defined for all numbers? How do you proove anything about their derivatives and general limits of them, when all you can resort to is a "drawing"?!

Maybe my worries are for nothing, but I still wanted to ask the question and ask whether there exists a definition of them purely in terms of algebraic means.

Rudin's book "Principles of Mathematical Analysis" defines sin(x) and cos(x) vie their Maclaurin series, then shows that series converge nicely for all x, that the functions have the derivatives they should, that sin(x)^2 + cos(x)^2 = 1 for all x, that sin(x) has a smallest positive zero (which we can call π), and that sin(x) and cos(x) are periodic of period 2π. All that can be done without any pictures at all---even without any geometry.

Of course, then you have the issue of connecting those functions to the usual trigonometric ones, so that you are allowed to use them in geometry. (Actually, I think Rudin denotes those functions as S(x) and C(x), and then shows that S and C have the properties of sin and cos.)

 

[Simon Bridge]

I heard that "sine" is the english form of an arabic word (after being latinized by monks and mangled on the way) which means "half chord".
Geometry is always defined on shapes, not arithmetic.
[didn't read all the way through]

You can note that there are arithmetic relations that will also get you there. They are just fancy ways of writing out the geometry.